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A cough can take a lot out of you!
You didn't mention if you treated it with anything? Also, prior to the cough was he ill? Sometimes a cough can linger for a long time after a respiratory infection.
Have his vaccines lasped? Whooping Cough in adults is fairly mild compared to its devastating effect on infants and children but it can still give some people a lenghty and pretty nasty cough.
It also could be caused by post nasal drip from an infection or from allergies. When he vomits, is it full of mucus? Or is it caused by the coughing itself; causes the gag reflex to be triggered? Or is there nausea?
Asthma can present as coughing.
He could also have some sort of obstruction that is irritating his airway and causing the cough.
If this has gone on for a month with no change its time to see a doctor.